Zernike Polynomials Research Articles

Effect of fixational eye movements in corneal topography measurements with optical coherence tomography.

There is an increasing interest in applying optical coherence tomography (OCT) to quantify the topography of ocular structures. However, in its most usual configuration, OCT data is acquired sequentially while a beam is scanned through the region of interest, and the presence of fixational eye movements can affect the accuracy of the technique. Several scan patterns and motion correction algorithms have been proposed to minimize this effect, but there is no consensus on the ideal parameters to obtain a correct topography. We have acquired corneal OCT images with raster and radial patterns, and modeled the data acquisition in the presence of eye movements. The simulations replicate the experimental variability in shape (radius of curvature and Zernike polynomials), corneal power, astigmatism, and calculated wavefront aberrations. The variability of the Zernike modes is highly dependent on the scan pattern, with higher variability in the direction of the slow scan axis. The model can be a useful tool to design motion correction algorithms and to determine the variability with different scan patterns.

Modeling and in-depth analysis of the mid-spatial-frequency error influenced by actual contact pressure distribution in sub-aperture polishing.

In ultra-precision optical processing, the sub-aperture polishing is prone to produce a mid-spatial-frequency (MSF) error. However, the generation mechanism of the MSF error is still not fully clarified, which seriously affects the further improvement of optical component performance. In this paper, it is proved that the actual contact pressure distribution between the workpiece and tool is a crucial source which affects the MSF error characteristics. A rotational periodic convolution (RPC) model is proposed to reveal the quantitative relationship among the contact pressure distribution, speed ratio (spin velocity/feed speed) and MSF error distribution. In-depth analyses show that the MSF error is linearly related to the symmetry level of contact pressure distribution and inversely proportional to the speed ratio, where the symmetry level is effectively evaluated by the proposed method based on Zernike polynomials. In the experiments, according to the actual contact pressure distribution obtained from the pressure-sensitive paper, the error rate of modeling results under different processing conditions is around 15%, which proves the validity of the proposed model. The influence of contact pressure distribution on the MSF error is further clarified through the establishment of RPC model, which can further promote the development of sub-aperture polishing.

Optical and Mechanical Performance and Feasibility Analysis of Meter-Level Corrector Lenses for Survey Telescope

A prime-focus optical system is a common type of large aperture survey telescope with a small obscuration ratio, high transmittance, and simple structure. As the detection capabilities and search efficiency of the telescope are improved, the aperture of the primary mirror becomes larger. The primary mirror aperture of a high-performance survey telescope has a four-meter scale, and the aperture of the corrector lenses of a corresponding prime-focus optical system can reach a one-meter scale. In this study, optical surface shape changes due to gravity were studied via an integrated optomechanical analysis. The Zernike polynomial was used to characterize the changed optical surface. The influence of optical surface variations on the wavefront and imaging quality of the optical system was analyzed by calculating the optical path differences. From the perspective of optical aberration, the optical properties of meter-level corrector lenses were quantitatively analyzed. The effect of the meter-level corrector lenses on imaging was explained in principle, and the feasibility of this optical scheme was verified.

Piezoelectric deformable mirror driven by unimorph actuator arrays on multi-spatial layers.

In order to reduce the cost of the piezo actuator array deformable mirror (DM), a piezoelectric DM driven by unimorph actuator arrays on multi-spatial layers is proposed. The actuator density can be multiplied by increasing the spatial layers of the actuator arrays. A low-cost DM prototype with 19 unimorph actuators located on three spatial layers is developed. The unimorph actuator can generate a wavefront deformation up to 11 µm at an operating voltage of 50 V. The DM can reconstruct typical low-order Zernike polynomial shapes accurately. The mirror can be flattened to 0.058 µm in RMS. Furthermore, a focal spot close to Airy spot is obtained in the far field after the aberrations of the adaptive optics testing system being corrected.

Phase aberration separation for holographic microscopy by alternating direction sparse optimization.

The morphology and dynamics of label-free tissues can be exploited by sample-induced changes in the optical field from quantitative phase imaging. Its sensitivity to subtle changes in the optical field makes the reconstructed phase susceptible to phase aberrations. We import variable sparse splitting framework on quantitative phase aberration extraction based on alternating direction aberration free method. The optimization and regularization in the reconstructed phase are decomposed into object terms and aberration terms. By formulating the aberration extraction as a convex quadratic problem, the background phase aberration can be fast and directly decomposed with the specific complete basis functions such as Zernike or standard polynomials. Faithful phase reconstruction can be obtained by eliminating global background phase aberration. The aberration-free two-dimensional and three-dimensional imaging experiments are demonstrated, showing the relaxation of the strict alignment requirements for the holographic microscopes.

Analysis and experiment of polarization characteristics of Off-axis freeform optical system

As freeform surfaces are widely used in off-axis optical systems with large apertures, large fields of view, and long focal lengths, the polarization effect caused by the system cannot be ignored. For this in mind, a polarization aberration analysis method for freeform optical system with fringe Zernike polynomial is proposed. An analytical model for the polarization aberration in this case is constructed. The influence of the surface shape on the polarization aberration is investigated. Our analysis shows that the distribution of phase aberration, diattenuation and retardance at the exit pupil introduced by the freeform surface in the system is related to the entrance pupil, incident field angle, and also closely dependent on the surface sag of the freeform surface. The introduction of freeform surfaces causes the system polarization aberration distribution to lose rotational symmetry. Finally, a freeform surface multi-angle polarization characteristic test platform was built to test the validity of the theoretical model. The polarization aberration characteristic data of the incident mirror at different angles of view at different positions were obtained and analyzed. The measured results show that the relative errors of phase aberration, diattenuation and retardance are all less than 7.2%, which is basically consistent with the theoretical simulation analysis results. The research in this paper provides a theoretical basis for guiding the design of an off-axis freeform surface polarization imaging optical system, and provides theoretical guidance for improving the polarization aberration calibration of a freeform surface optical system.

Compensation machining of freeform based on fast tool servo system.

With the advantages of large field of view, low cost and simple structure, the freeform optical system has extensive requirements in space exploration and other fields. However, the current machining methods for freeform are difficult to meet the requirements of optical use. Based on a developed fast servo tool (FTS) device, this paper proposes an error compensation turning method for freeforms. Firstly, the Zernike polynomial fitting method is used to reconstruct the freeform surface shape error obtained from off-line measurement, and the offset compensation is used to correct the tool path. Then, the compensation processing physical system is built to simulate the off-line compensation processing of the workpiece to verify the feasibility of compensation processing. Finally, the turning compensation processing of convex freeform aluminum mirror is carried out, and the surface accuracy of the workpiece meets the requirements of visible band. The research results have important practical significance for realizing the fast response machining of free-form surface mirror.

Simulation of the Human Myopic Eye Cornea Compensation Based on the Analysis of Aberrometric Data.

Various diffractive, refractive and holographic optical elements, such as diffraction gratings; microlens raster; phase plates; multi-order diffractive optical elements; adaptive mirrors; diffractive and refractive axicons; holographic multiplexes and many others are used to analyze wavefront aberrations. We shortly discuss the features (advantages and disadvantages) of various wavefront aberration sensors in the Introduction. The main part of the paper is devoted to the analysis of the weight coefficients of Zernike polynomials obtained during medical examinations of the cornea in the human eye. Using data obtained by aberrometers, the average values of the Zernike polynomial coefficients for the anterior and posterior surfaces of the healthy eye cornea and a myopic one were calculated. The original wavefront for the anterior and posterior surfaces of the cornea was restored separately, as well as the total wave aberration. For an objective assessment of the quality of vision, the corresponding point spread functions (PSFs) were calculated. We propose to compensate for the aberrations of the myopic eye, taking into account the physical features of the corneal surface. The results of numerical simulation showed that in order to improve the quality of the patient's vision, it is necessary to take into account high-order aberrations of the anterior surface of the cornea in the form of a coma of the third order and aberrations of the fourth order.

Quantifying the morphology of eyeballs with posterior staphyloma with Zernike polynomials.

Purpose: To quantify the morphology of eyeballs with posterior staphyloma (PS) with Zernike decomposition and to explore the association between Zernike coefficients with existing PS classification. Methods: Fifty-three eyes with high myopia (HM, ≤-6.00D) and 30 with PS were included. PS was classified with traditional methods based on OCT findings. Eyeballs' morphology was obtained with 3D MRI, from which the height map of the posterior surface was extracted. Zernike decomposition was performed to derive the coefficients of the 1st-27th items, which were compared between HM and PS eyes with the Mann-Whitney-U test. Receiver operating characteristics (ROC) analysis was used to test the effectiveness of using Zernike coefficients to discriminate PS from HM. Results: Compared to HM eyeballs, PS eyeballs had significantly increased vertical and horizontal tilt, oblique astigmatism, defocus, vertical and horizontal coma, and higher order aberrations (HOA) (all Ps < 0.05). HOA was the most effective in PS classification with an area under the ROC curve (AUROC) value of 0.977. Among the 30PS, 19 were the wide macular type with large defocus and negative spherical aberration; 4 were the narrow macular type with positive spherical aberration; 3 were inferior PS with greater vertical tilt, and 4were peripapillary PS with larger horizontal tilt. Conclusion: PS eyes have significantly increased Zernike coefficients, and HOA is the most effective parameter to differentiate PS from HM. The geometrical meaning of the Zernike components showed great accordance with PS classification.

Typical localised element-specific finite element anterior eye model

PurposeThe study presents an averaged anterior eye geometry model combined with a localised material model that is straightforward, appropriate and amenable for implementation in finite element (FE) modelling. MethodsBoth right and left eye profile data of 118 subjects (63 females and 55 males) aged 22–67 years (38.5 ± 7.6) were used to build an averaged geometry model. Parametric representation of the averaged geometry model was achieved through two polynomials dividing the eye into three smoothly connected volumes. This study utilised the collagen microstructure x-ray data of 6 ex-vivo healthy human eyes, 3 right eyes and 3 left eyes in pairs from 3 donors, 1 male and 2 females aged between 60 and 80 years, to build a localised element-specific material model for the eye. ResultsFitting the cornea and the posterior sclera sections to a 5th-order Zernike polynomial resulted in 21 coefficients. The averaged anterior eye geometry model recorded a limbus tangent angle of 37° at a radius of 6.6 mm from the corneal apex. In terms of material models, the difference between the stresses generated in the inflation simulation up to 15 mmHg in the ring-segmented material model and localised element-specific material model were significantly different (p < 0.001) with the ring-segmented material model recording average Von-Mises stress 0.0168 ± 0.0046 MPa and the localised element-specific material model recording average Von-Mises stress 0.0144 ± 0.0025 MPa. ConclusionsThe study illustrates an averaged geometry model of the anterior human eye that is easy to generate through two parametric equations. This model is combined with a localised material model that can be used either parametrically through a Zernike fitted polynomial or non-parametrically as a function of the azimuth angle and the elevation angle of the eye globe. Both averaged geometry and localised material models were built in a way that makes them easy to implement in FE analysis without additional computation cost compared to the limbal discontinuity so-called idealised eye geometry model or ring-segmented material model.

Intraocular composition of higher order aberrations in non-myopic children.

This study examined anterior corneal, internal ocular, and total ocular higher order aberrations (HOA's), and retinal image quality in a non-myopic, paediatric cohort. Anterior corneal aberrations were derived from corneal topography data captured using a Placido disk videokeratoscope (E300, Medmont International), and whole eye HOA's were measured using a Hartmann-Shack wavefront sensor (COAS-HD, Wavefront Sciences). The associations between HOA's and age, sex, refractive error, and axial length were explored using correlation analyses. Data for 84 children aged between 5 and 12 years (mean ± standard deviation spherical equivalent refraction (SER), +0.63 ± 0.35 D; range 0.00 to +1.75 D) were included, and an eighth order Zernike polynomial was fit for 4 and 6 mm pupil diameters for both the anterior corneal and total ocular HOA's, from which internal ocular HOA's were calculated via subtraction following alignment to a common reference axis (pupil centre). Internal ocular HOA's were of greater magnitude than previous studies of adolescents and adults, however partial internal "compensation" of HOA's was observed, which resulted in reduced levels of HOA's and excellent retinal image quality. Few significant associations were observed between HOA's and age, SER, and axial length (all correlations, p > 0.001), and there were minimal sex-based differences (all comparisons, p > 0.005). Coefficients for vertical coma ( and ) and spherical aberration ( and ), were most strongly associated with the visual Strehl ratio based on the optical transfer function (VSOTF), which indicated that the absolute magnitudes of these Zernike coefficients have the greatest impact on retinal image quality in this paediatric cohort. These findings provide an improved understanding of the optics and retinal image quality of children's eyes.

Exact formula to relate optical path differences and transversal aberration components.

An exact equation to relate the optical path differences (OPD) with its transversal aberration components (TAC) is determined. The OPD-TAC equation reproduces the Rayces formula and introduces the coefficient for the longitudinal aberration. The defocus orthonormal Zernike polynomial (Z DF) is not a solution for the OPD-TAC equation since the obtained longitudinal defocus depends on the ray height on the exit pupil, meaning that it cannot be interpreted as a defocus. In order to find an exact expression for OPD defocus, first, a general relationship between the wavefront shape and its OPD is established. Second, an exact formula for the defocus OPD is established. Finally, it is proved that only the exact defocus OPD is an exact solution of the exact OPD-TAC equation.

Evaluation method for the design results of space gravitational-wave telescopes

Unlike traditional imaging telescopes, space gravitational-wave telescope has higher requirements for wavefront error and must meet the stability requirement of tilt-to-length (TTL). The tolerance analysis results of current telescopes can judge whether the wavefront error meets the requirements, but they cannot be used as the judgment result of the TTL noise coupling coefficient of the telescope. To analyze the influence of manufacturing errors of the telescope on the coupling coefficient of TTL noise, based on the error amount and the Zernike polynomial first-order sensitivity matrix, we established the sensitivity matrix of the mapping relationship between telescope manufacturing errors and coupling coefficients. For example, consider the design results of an off-axis four-mirror space gravitational-wave telescope. If the wavefront error was used as the judgment result to determine the tolerance distribution, the cumulative probability of the coupling coefficient meeting the requirements would be 66.7%. Furthermore, using the coupling coefficient as the control requirement and determining the tolerance allocation results according to the coupling coefficient sensitivity matrix, the cumulative probability increased to 93.8%. The necessity of using the coupling coefficient as an evaluation method for the design results of gravitational-wave telescopes was verified. This evaluation method can provide meaningful guidance for the design results of gravitational-wave telescopes.

A Review on Zernike Coefficient-Solving Algorithms (CSAs) Used for Integrated Optomechanical Analysis (IOA)

An integrated optomechanical analysis (IOA) can predict the response of an optomechanical system to temperature, gravity, vibrations, and other local loadings; thus, the normal operation of instruments under special conditions is guaranteed. Zernike polynomials are the most popular for fitting the IOA-derived mechanical deformation data. By solving the Zernike coefficients of all deformed optical surfaces, the relationship between aberrations and deformations can be further revealed. The process of IOA is summarized in this article. The principles of four primary Zernike coefficient-solving algorithms (CSAs) were expounded, and the corresponding applications are reviewed in detail, including the least squares method, the Gram–Schmidt orthogonalized method, the Householder transformation, and singular value decomposition (SVD). Artificial neural networks (ANNs) trained for solving a similar overdetermined set of equations are also discussed; an innovative Zernike CSA based on a one-dimensional convolutional neural network (1D-CNN) was proposed, emphasizing its potential for Zernike CSA. The feasibility of the neural network method was verified by conducting experiments on the primary mirror of the front reflection system of a space camera. This review can provide references for the precise optimization of IOA.

Insensitivity of partially coherent Gaussian -Schell model beams to the impact of dynamic Kolmogorov type turbulence

ABSTRACT The effect of dynamic Kolmogorov kind of turbulence on Gaussian- Schell model beams in different coherence regimes have been explored in detail. The propagation characteristics of such turbulence impacted beams are quantitatively verified at the laboratory level by calculating beam wandering, scintillation index, and Zernike polynomials. It has been experimentally verified that Gaussian Schell model beams are more resilient to the impact of turbulence compared to their fully coherent counterparts under different turbulence strengths. These results find applications in free-space optical communications.

Low-wavenumber compensation with Zernike tilt for non-Kolmogorov turbulence phase screens.

The fast-Fourier-transform-based filtering method for phase screen generation remains popular for numerical simulation of optical propagation through turbulence; however, these screens inherently underrepresent the spectral density at low wavenumbers. Here, the "Z-tilt" approach is explored to augment the spectral density at low wavenumbers by adding a random phase tilt, which is derived from the wavefront phase statistics of a Zernike polynomial basis. This approach is computationally efficient and can be applied to any statistically homogeneous and isotropic refractive index field. An analytic result is provided for the von Kármán spectrum with finite outer scale. In a quantitative comparison with phase screens compensated for using a common subharmonic approach, the Z-tilt method shows the best agreement with the analytical structure function when the outer scale is greater than about three times the screen dimension. For outer scales of the order of the screen dimension, the subharmonic and a modified Z-tilt method give the most accurate results. A propagation simulation demonstrates that the aperture-averaged angle-of-arrival variance is accurately predicted using the Z-tilt method.

Wavelength-shifting interferometry using the frequency-modulated Chirp-Z transform and effective multi-harmonic sampling

In wavelength-shifting measurements of the transparent flat, multi-surface harmonics of the measured surfaces with different frequencies provide the corresponding intensity information in the captured interferograms. The limitations of such measurements arise from the limited measurement distance and unsatisfactory frequency detection accuracy. Therefore, a local spectral intensifying method using the Chirp-Z transform is introduced. Based on the distribution characteristics of the phase-shifts within each harmonic period, the calculation method of the measurement distance and the available sampling frequency that can avoid under-sampling is proposed. Based on the above methods, efficient multi-surface phase demodulation can be implemented. Using the Zernike polynomials, the robustness of the proposed algorithm to some common measurement errors is analyzed and demonstrated. In experiments, two transparent plates with thicknesses of 6.35 mm and 15 mm were measured using a wavelength-shifting interferometer. The RMS errors of three repeatability measurements in the experiments are 5.71 nm and 0.64 nm, respectively. Both the simulation analysis and the experiments verify the effectiveness of the proposed method.

Research on thermal stability of monolithic near-infrared Doppler asymmetric spatial heterodyne interferometer

A Doppler asymmetric spatial heterodyne (DASH) interferometer was designed to measure atmospheric winds at a height of 60 to 80 km by observing the airglow emission line of molecular oxygen at 867 nm. The designed monolithic DASH interferometer exhibited decent thermal stability. The phase thermal drift of the fabricated interferometer obtained from thermal performance measurements was 0.376 rad / ° C. To accurately model and minimize the thermal drift performance of an interferometer in the design phase, it is necessary to include the influence of thermal distortion of the monolithic interferometer components. Therefore, an optical–structural–thermal integrated analysis method based on Zernike polynomials was proposed to accurately calculate the phase thermal drift of the interferometer. The optical model modified by the finite-element method calculated the phase thermal drift to be 0.420 rad / ° C, which agreed with the experimental result within 11.7%. This analysis method can accurately calculate and optimize thermal stability during the design of a DASH interferometer.

Estimating conformational landscapes from Cryo-EM particles by 3D Zernike polynomials

The new developments in Cryo-EM Single Particle Analysis are helping us to understand how the macromolecular structure and function meet to drive biological processes. By capturing many states at the particle level, it is possible to address how macromolecules explore different conformations, information that is classically extracted through 3D classification. However, the limitations of classical approaches prevent us from fully understanding the complete conformational landscape due to the reduced number of discrete states accurately reconstructed. To characterize the whole structural spectrum of a macromolecule, we propose an extension of our Zernike3D approach, able to extract per-image continuous flexibility information directly from a particle dataset. Also, our method can be seamlessly applied to images, maps or atomic models, opening integrative possibilities. Furthermore, we introduce the ZART reconstruction algorithm, which considers the Zernike3D deformation fields to revert particle conformational changes during the reconstruction process, thus minimizing the blurring induced by molecular motions.

Separate the primary aberration for the deformation error of optical flat in 300-mm-aperture vertical Fizeau interferometer

As a key component in a phase-shifting interferometer, the optical flat is often affected by supporting methods and generates deformation. In this paper, we propose a method to separate the deformation error through rotation measurement and Zernike polynomial fitting. We studied the principles, simulations, and experiments of this method. First, a support model is established in Comsol, and the deformation of the optical flat is analyzed. Then, the numerical simulation of the separation method for the deformation is performed, and the corresponding experiment is carried out on a Φ300 mm vertical Fizeau interferometer to verify the correctness of the method. The results show that numerical calculation and experimental measurement results are in good agreement, the difference between the PV value and the RMS value are only 0.004λ and 0.002λ, respectively. Finally, the error term that affected the calculation accuracy was analyzed. This proves that the proposed method can effectively separate the deformation introduced by the support.